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cleartomark
%%EndFont
%%BeginFont: CMSS12
%!PS-AdobeFont-1.1: CMSS12 1.0
%%CreationDate: 1991 Aug 20 17:33:47
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSS12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMSS12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 77 /M put
dup 97 /a put
dup 105 /i put
dup 114 /r put
dup 118 /v put
dup 121 /y put
readonly def
/FontBBox{-62 -251 978 758}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMR17
%!PS-AdobeFont-1.1: CMR17 1.0
%%CreationDate: 1991 Aug 20 16:38:24
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR17) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR17 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 40 /parenleft put
dup 41 /parenright put
readonly def
/FontBBox{-33 -250 945 749}readonly def
currentdict end
currentfile eexec
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2BDBF16FBC7512FAA308A093FE5F075EA0A10A15B0ED05D5039DA41B32B16E95
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cleartomark
%%EndFont
%%BeginFont: CMMI12
%!PS-AdobeFont-1.1: CMMI12 1.100
%%CreationDate: 1996 Jul 27 08:57:55
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 12 /beta put
dup 13 /gamma put
dup 14 /delta put
dup 16 /zeta put
dup 22 /mu put
dup 23 /nu put
dup 25 /pi put
dup 26 /rho put
dup 34 /epsilon put
dup 58 /period put
dup 59 /comma put
dup 60 /less put
dup 61 /slash put
dup 62 /greater put
dup 64 /partialdiff put
dup 66 /B put
dup 67 /C put
dup 68 /D put
dup 69 /E put
dup 70 /F put
dup 71 /G put
dup 75 /K put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 79 /O put
dup 82 /R put
dup 85 /U put
dup 86 /V put
dup 88 /X put
dup 90 /Z put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 113 /q put
dup 114 /r put
dup 115 /s put
dup 118 /v put
dup 120 /x put
dup 121 /y put
dup 122 /z put
readonly def
/FontBBox{-30 -250 1026 750}readonly def
currentdict end
currentfile eexec
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%%EndFont
%%BeginFont: CMBX12
%!PS-AdobeFont-1.1: CMBX12 1.0
%%CreationDate: 1991 Aug 20 16:34:54
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMBX12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
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end readonly def
/FontName /CMBX12 def
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/FontMatrix [0.001 0 0 0.001 0 0] readonly def
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readonly def
/FontBBox{-53 -251 1139 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMMIB10
%!PS-AdobeFont-1.1: CMMIB10 1.100
%%CreationDate: 1996 Jul 23 07:54:00
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMIB10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMIB10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 97 /a put
dup 120 /x put
dup 121 /y put
dup 122 /z put
readonly def
/FontBBox{-15 -250 1216 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMMIB7
%!PS-AdobeFont-1.1: CMMIB7 001.100
%%CreationDate: 1996 Jul 27 07:35:50
% Computer Modern fonts were designed by Donald E. Knuth.
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (001.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMIB7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMMIB7 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 120 /x put
readonly def
/FontBBox{0 -250 1294 750}readonly def
currentdict end
currentfile eexec
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0CBDA9FF1FC99217BB643C87DD94CA2BB185BCD37C128178A0F799AAAC30DBEB
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cleartomark
%%EndFont
%%BeginFont: MSAM10
%!PS-AdobeFont-1.1: MSAM10 2.1
%%CreationDate: 1993 Sep 17 09:05:00
% Math Symbol fonts were designed by the American Mathematical Society.
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (2.1) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (MSAM10) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /MSAM10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 4 /squaresolid put
readonly def
/FontBBox{8 -463 1331 1003}readonly def
currentdict end
currentfile eexec
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0A8624A495F142A3D4BEDF75EA6AEB3F283B1B527748949E4767368AB7391F22
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7D54E7688CE550D5F283BAA71FECD03B45052F6C5467A3A93B1B1AF15B611A31
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5D0A41A7321E0B76A1DC8D3129CA0FE67228C0D9B4A56B3D6551D1B34ADBB2B9
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cleartomark
%%EndFont
%%BeginFont: CMSY6
%!PS-AdobeFont-1.1: CMSY6 1.0
%%CreationDate: 1991 Aug 15 07:21:34
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY6 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /minus put
dup 3 /asteriskmath put
dup 49 /infinity put
readonly def
/FontBBox{-4 -948 1329 786}readonly def
currentdict end
currentfile eexec
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39CC36BC3F9D9A3148799CC266A967ABDE78EA4FB6FEA0D136483C8598000FE3
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4FEFDD6F96C705878ECB5EC51D863BE49A47488C63F3125870406EFAC20B9CF7
11AF72C2A69F279973D8FD125AF0BB43D630E0686EBB4AB2092466A3FEBE36F5
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83EAE5CB476157BBC570FBE1757524A58E4C126A04F304CABF91D5B338721E3A
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cleartomark
%%EndFont
%%BeginFont: MSBM7
%!PS-AdobeFont-1.1: MSBM7 2.1
%%CreationDate: 1992 Oct 17 08:30:50
% Math Symbol fonts were designed by the American Mathematical Society.
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (2.1) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (MSBM7) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /MSBM7 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 78 /N put
dup 82 /R put
readonly def
/FontBBox{0 -504 2615 1004}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMSS8
%!PS-AdobeFont-1.1: CMSS8 1.0
%%CreationDate: 1991 Aug 20 17:34:12
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSS8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMSS8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 77 /M put
dup 97 /a put
dup 105 /i put
dup 114 /r put
dup 118 /v put
dup 121 /y put
readonly def
/FontBBox{-65 -250 1062 761}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMR6
%!PS-AdobeFont-1.1: CMR6 1.0
%%CreationDate: 1991 Aug 20 16:39:02
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR6 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 3 /Lambda put
dup 40 /parenleft put
dup 41 /parenright put
dup 43 /plus put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 58 /colon put
dup 61 /equal put
dup 99 /c put
dup 100 /d put
dup 104 /h put
dup 112 /p put
readonly def
/FontBBox{-20 -250 1193 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMMI6
%!PS-AdobeFont-1.1: CMMI6 1.100
%%CreationDate: 1996 Jul 23 07:53:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI6 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 13 /gamma put
dup 58 /period put
dup 59 /comma put
dup 78 /N put
dup 88 /X put
dup 100 /d put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 110 /n put
dup 113 /q put
dup 114 /r put
dup 115 /s put
readonly def
/FontBBox{11 -250 1241 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMSY8
%!PS-AdobeFont-1.1: CMSY8 1.0
%%CreationDate: 1991 Aug 15 07:22:10
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /minus put
dup 3 /asteriskmath put
dup 6 /plusminus put
dup 20 /lessequal put
dup 21 /greaterequal put
dup 26 /propersubset put
dup 33 /arrowright put
dup 48 /prime put
dup 49 /infinity put
dup 50 /element put
dup 54 /negationslash put
dup 69 /E put
dup 70 /F put
dup 71 /G put
dup 92 /intersection put
dup 102 /braceleft put
dup 103 /braceright put
dup 106 /bar put
readonly def
/FontBBox{-30 -955 1185 779}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMR8
%!PS-AdobeFont-1.1: CMR8 1.0
%%CreationDate: 1991 Aug 20 16:39:40
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /Gamma put
dup 3 /Lambda put
dup 33 /exclam put
dup 40 /parenleft put
dup 41 /parenright put
dup 43 /plus put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 58 /colon put
dup 61 /equal put
dup 94 /circumflex put
dup 97 /a put
dup 99 /c put
dup 100 /d put
dup 104 /h put
dup 105 /i put
dup 108 /l put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 118 /v put
readonly def
/FontBBox{-36 -250 1070 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: MSBM10
%!PS-AdobeFont-1.1: MSBM10 2.1
%%CreationDate: 1993 Sep 17 11:10:37
% Math Symbol fonts were designed by the American Mathematical Society.
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (2.1) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (MSBM10) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /MSBM10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 78 /N put
dup 82 /R put
dup 90 /Z put
readonly def
/FontBBox{-55 -420 2343 920}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMSY10
%!PS-AdobeFont-1.1: CMSY10 1.0
%%CreationDate: 1991 Aug 15 07:20:57
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /minus put
dup 1 /periodcentered put
dup 2 /multiply put
dup 20 /lessequal put
dup 21 /greaterequal put
dup 24 /similar put
dup 25 /approxequal put
dup 26 /propersubset put
dup 28 /lessmuch put
dup 33 /arrowright put
dup 38 /arrowsoutheast put
dup 41 /arrowdblright put
dup 44 /arrowdblboth put
dup 49 /infinity put
dup 50 /element put
dup 51 /owner put
dup 54 /negationslash put
dup 55 /mapsto put
dup 56 /universal put
dup 59 /emptyset put
dup 69 /E put
dup 70 /F put
dup 71 /G put
dup 91 /union put
dup 92 /intersection put
dup 102 /braceleft put
dup 103 /braceright put
dup 106 /bar put
dup 110 /backslash put
readonly def
/FontBBox{-29 -960 1116 775}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMMI8
%!PS-AdobeFont-1.1: CMMI8 1.100
%%CreationDate: 1996 Jul 23 07:53:54
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 12 /beta put
dup 13 /gamma put
dup 22 /mu put
dup 23 /nu put
dup 26 /rho put
dup 34 /epsilon put
dup 58 /period put
dup 59 /comma put
dup 60 /less put
dup 61 /slash put
dup 62 /greater put
dup 66 /B put
dup 67 /C put
dup 75 /K put
dup 78 /N put
dup 85 /U put
dup 88 /X put
dup 100 /d put
dup 101 /e put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 113 /q put
dup 114 /r put
dup 115 /s put
dup 118 /v put
dup 120 /x put
dup 122 /z put
readonly def
/FontBBox{-24 -250 1110 750}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMEX10
%!PS-AdobeFont-1.1: CMEX10 1.00
%%CreationDate: 1992 Jul 23 21:22:48
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMEX10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMEX10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /parenleftbig put
dup 1 /parenrightbig put
dup 2 /bracketleftbig put
dup 3 /bracketrightbig put
dup 8 /braceleftbig put
dup 9 /bracerightbig put
dup 12 /vextendsingle put
dup 16 /parenleftBig put
dup 17 /parenrightBig put
dup 18 /parenleftbigg put
dup 19 /parenrightbigg put
dup 40 /braceleftBigg put
dup 48 /parenlefttp put
dup 49 /parenrighttp put
dup 50 /bracketlefttp put
dup 51 /bracketrighttp put
dup 52 /bracketleftbt put
dup 53 /bracketrightbt put
dup 64 /parenleftbt put
dup 65 /parenrightbt put
dup 73 /contintegraldisplay put
dup 80 /summationtext put
dup 81 /producttext put
dup 82 /integraltext put
dup 88 /summationdisplay put
dup 89 /productdisplay put
dup 90 /integraldisplay put
dup 91 /uniondisplay put
dup 104 /bracketleftBig put
dup 105 /bracketrightBig put
dup 110 /braceleftBig put
dup 111 /bracerightBig put
readonly def
/FontBBox{-24 -2960 1454 772}readonly def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMR12
%!PS-AdobeFont-1.1: CMR12 1.0
%%CreationDate: 1991 Aug 20 16:38:05
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /Gamma put
dup 1 /Delta put
dup 3 /Lambda put
dup 22 /macron put
dup 33 /exclam put
dup 40 /parenleft put
dup 41 /parenright put
dup 43 /plus put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 58 /colon put
dup 61 /equal put
dup 91 /bracketleft put
dup 93 /bracketright put
dup 94 /circumflex put
dup 97 /a put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 105 /i put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 120 /x put
dup 126 /tilde put
readonly def
/FontBBox{-34 -251 988 750}readonly def
currentdict end
currentfile eexec
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3567 y
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3203 3837 a 0 TeXcolorgray -2 x FD(20)p
0 TeXcolorgray 3296 3774 a
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3296 3774 a 3296 3835 a
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/Link/Dest (cite.wr) cvn H.B /ANN pdfmark end
3296
3835 a FD(],)27 b(see)g(the)109 3955 y(re)m(vie)n(w)e([)397
3957 y
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397 3957 a 0 TeXcolorgray -2 x FD(18)p 0 TeXcolorgray
490 3895 a
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490 3895 a 490 3955 a
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/Link/Dest (cite.samaj-wr) cvn H.B /ANN pdfmark end
490 3955 a FD(].)e(Consider)h(par)s
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4121 y(\014)109 4338 y FD(wher)27 b Fy(B)5 b Fx(\()p
Fy(x;)17 b Fx(1\))26 b FD(is)h(the)f(ball)g(of)g(r)o(adius)h
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g(it)g(quali\002es)i(ne)m(v)n(er)s(theless)g(as)f(an)g(attr)o(activ)n
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y Fq(1)213 4889 y Fw(X)208 5100 y Fv(N)7 b Fr(=0)388
4916 y Fy(z)437 4880 y Fv(N)p 388 4961 117 4 v 389 5052
a Fy(N)j Fx(!)532 4848 y Fw(Z)587 5074 y Fr(\003)636
5055 y Fp(N)714 4984 y Fy(e)759 4942 y Fq(\000)p Fv(\014)s(U)905
4953 y Fo(\003)951 4942 y Fr(\()p Fi(x)q Fr(\))1056 4984
y Fx(d)p Fh(x)1114 5307 y Fx(=)27 b Fy(e)1262 5266 y
Fq(\000)p Fv(z)1350 5275 y Fo(2)1385 5266 y Fq(j)p Fr(\003)p
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5432 y Fo(1)1584 5423 y Fv(;N)1660 5432 y Fo(2)1694 5423
y Fr(=0)1811 5239 y Fy(z)1860 5196 y Fv(N)1916 5205 y
Fo(1)1856 5264 y Fr(1)p 1811 5284 145 4 v 1811 5375 a
Fy(N)1889 5390 y Fr(1)1929 5375 y Fx(!)1976 5239 y Fy(z)2025
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FD(Again,)23 b(this)h(matches)f(Corollar)s(y)1236 1025
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1354 1023 a FD(.)109
1218 y(A)32 b(second)h(consequence)h(of)d(Theorem)1511
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1629 1218 a 31 w
FD(is)i(that,)d(e)m(v)n(en)j(when)f(there)g(is)g(no)g(phase)h(tr)o
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(f)m(ast)109 1820 y(increase)j(from)f(small)g(to)g(large)h(density)-8
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(ollo)o(wing)f(conjecture:)109 2126 y
SDict begin H.S end
109 2126 a 109
2126 a
SDict begin 14.5 H.A end
109 2126 a 109 2126 a
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3652 2282 a 109 2403
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321 2403
a FD(.)g(Then)1304 2523 y
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b(that)g(f)m(or)h(pair)g(potentials)f(whose)i(\002nite)e(par)s(t)h(is)g
(integ)o(r)o(ab)n(le)o(,)g(w)o(e)g(ha)n(v)n(e)g(the)g(lo)o(w)o(er)g
(bound)g(Eq.)g(\()3413 2780 y
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FD(11)p 0 TeXcolorgray 3507 2720 a
SDict begin H.R end
3507 2720 a 3507 2780
a
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/Link/Dest (equation.2.11) cvn H.B /ANN pdfmark end
3507 2780 a FD(\))g(on)109 2901 y(the)h(liminf)m(.)f(Hence)i(the)f
(only)g(par)s(t)h(that)f(is)g(open)h(in)f(the)g(pre)m(vious)i
(conjecture)f(is)g(an)f(upper)h(bound)g(on)f(the)109
3021 y(limsup)m(.)109 3189 y(In)f(f)m(act,)g(if)g(w)o(e)h(are)g
(willing)f(to)g(allo)o(w)h(f)m(or)g(a)f(minor)h(modi\002cation)f(of)g
(the)h(de\002nition)f(of)g Fy(R)2959 3153 y Fn(Ma)n(y)3090
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(Theorem)1950 3311 y
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SDict begin H.R end
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3309 a
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2068 3309 a 24 w FD(with)g(the)h(bounds)g(from)f([)2930
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/Link/Dest (cite.penrose) cvn H.B /ANN pdfmark end
3023 3309 a FD(].)g(First)g(recall)
h(that)109 3430 y(the)f(pressure)j Fy(\014)6 b(p)716
3445 y Fr(\003)768 3430 y Fx(\()p Fy(\014)g(;)17 b(z)t
Fx(\))p FD(,)25 b(de\002ned)g(via)g(the)g(logar)q(ithm)f(of)h(the)g(g)o
(r)o(and-canonical)h(par)s(tition)f(function)f(in)h(a)109
3550 y(\002nite)e(bo)m(x)i Fx(\003)i(=)h([0)p Fy(;)17
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SDict begin H.S end
2403
3550 a 0 TeXcolorgray FD(4)p 0 TeXcolorgray 2450 3490
a
SDict begin H.R end
2450 3490 a 2450 3550 a
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/Link/Dest (equation.2.4) cvn H.B /ANN pdfmark end
2450 3550 a FD(\),)f(with)g(v)n
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b(sup)1561 3694 y Fq(j)p Fr(\003)p Fq(j!1)1811 3670 y
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3634 y Fn(Ma)n(y)2493 3670 y Fx(\()p Fy(\014)6 b Fx(\))p
FD(.)109 3711 y
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109 3711 a 109 3711 a
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4237 y Fq(j)p Fr(\003)p Fq(j!1)1033 4152 y Fy(\014)1094
4110 y Fq(\000)p Fr(1)1205 4152 y Fx(log)g Fy(R)1422
4104 y Fn(Ma)n(y)1421 4179 y Fr(\003)1554 4152 y Fx(\()p
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Fr(\003)p Fq(j!1)2309 4152 y Fy(\014)2370 4110 y Fq(\000)p
Fr(1)2480 4152 y Fx(log)h Fy(R)2698 4104 y Fn(Ma)n(y)2697
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(might)g(be)g(\223chemical\224)h(tr)o(ansitions)g([)2333
4570 y
SDict begin H.S end
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SDict begin H.R end
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b(.)109 2533 y
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2981 82 4 v 1045 3073 a Fy(k)s Fx(!)1182 2910 y Fw(X)1153
3119 y Fv(\015)h Ff(conn.)1372 2869 y Fw(Z)1427 3094
y Fr(\()p Fl(R)1502 3075 y Fp(d)1539 3094 y Fr(\))1566
3075 y Fp(k)q Fk(\000)p Fo(1)1739 2910 y Fw(Y)1703 3125
y Fr(\()p Fv(ij)t Fr(\))p Fq(2)p Fv(\015)1919 3004 y
Fy(f)1967 3019 y Fv(ij)2027 3004 y Fx(\()p Fh(x)p Fx(\)d)p
Fy(x)2278 3019 y Fr(2)2334 3004 y Ft(\001)17 b(\001)g(\001)e
Fx(d)p Fy(x)2576 3019 y Fv(k)2620 3004 y Fy(;)101 b(x)2803
3019 y Fr(1)2870 3004 y Fx(:=)28 b(0)p Fy(:)424 b FD(\(23\))109
3317 y(The)22 b(sum)h(is)g(o)o(v)n(er)g(connected,)h(undirected)f(g)o
(r)o(aphs)h Fy(\015)33 b Fx(=)27 b(\()p Fy(V)5 b(;)17
b(E)6 b Fx(\))23 b FD(with)f(v)n(er)s(tices)i Fx(1)p
Fy(;)17 b(:)g(:)g(:)f(;)h(k)s FD(,)22 b(and)3355 3242
y Fw(Q)3449 3346 y Fr(\()p Fv(ij)t Fr(\))p Fq(2)p Fv(\015)109
3445 y FD(is)h(the)h(product)g(o)o(v)n(er)f(edges)i Ft(f)p
Fy(i;)17 b(j)6 b Ft(g)27 b(2)h Fy(E)6 b FD(,)23 b Fy(i)28
b(<)g(j)h FD(\(no)24 b(self-edges)g Fx(\()p Fy(ii)p Fx(\))p
FD(\).)109 3613 y(Let)f(us)h(star)s(t)g(with)f(a)g(look)h(at)f(the)g
Fy(\014)34 b Ft(!)27 b(1)c FD(beha)n(vior)i(f)m(or)f(an)f(individual)h
(g)o(r)o(aph.)g(Obser)s(ving)g(that)795 3897 y Fy(f)843
3912 y Fv(ij)903 3897 y Fx(\()p Fh(x)p Fx(\))k(=)1176
3726 y Fw(\()1256 3828 y Fx(\(1)22 b(+)g Fy(o)p Fx(\(1\)\))17
b(exp\()p Ft(\000)p Fy(\014)6 b(v)2061 3843 y Fv(ij)2122
3828 y Fx(\()p Fh(x)p Fx(\)\))p Fy(;)169 b(v)2545 3843
y Fv(ij)2606 3828 y Fx(\()p Fh(x)o Fx(\))28 b Fy(<)f
Fx(0)p Fy(;)1256 3973 y Ft(\000)p Fx(1)c(+)f Fy(o)p Fx(\(1\))p
Fy(;)796 b(v)2545 3988 y Fv(ij)2606 3973 y Fx(\()p Fh(x)o
Fx(\))28 b Ft(\025)g Fx(0)p Fy(;)109 4168 y FD(w)o(e)c(get)639
4215 y Fw(Y)603 4431 y Fr(\()p Fv(ij)t Fr(\))p Fq(2)p
Fv(\015)802 4225 y Fw(\014)802 4285 y(\014)835 4310 y
Fy(f)883 4325 y Fv(ij)943 4310 y Fx(\()p Fh(x)p Fx(\))1085
4225 y Fw(\014)1085 4285 y(\014)1146 4310 y Fx(=)j(\(1)22
b(+)g Fy(o)p Fx(\(1\)\))17 b(exp)1831 4199 y Fw(\020)1891
4310 y Ft(\000)p Fy(\014)2073 4215 y Fw(X)2046 4431 y
Fr(\()p Fv(ij)t Fr(\))p Fq(2)p Fv(\015)2261 4310 y Fy(v)2308
4325 y Fv(ij)2368 4310 y Fx(\()p Fh(x)p Fx(\))p Fg(1)p
Fx(\()p Fy(v)2651 4325 y Fv(ij)2712 4310 y Fx(\()p Fh(x)o
Fx(\))28 b Fy(<)f Fx(0\))3071 4199 y Fw(\021)3131 4310
y Fy(:)109 4573 y FD(In)i(the)g(e)m(xponent,)h(only)g(negativ)n(e)g
(inter)o(actions)g(appear)l(.)g(As)g(a)g(result,)f(w)o(e)h(ma)m(y)g
(end)g(up)g(with)f(energies)109 4693 y(m)o(uch)d(smaller)g(than)h(the)f
(g)o(round)h(state)f(energy)-8 b(,)26 b(seemingly)g(contr)o(adicting)h
(Theorem)3053 4695 y
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3053 4695 a 0 TeXcolorgray -2 x
FD(3.1)p 0 TeXcolorgray 3171 4633 a
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3171 4633 a 3171
4693 a
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3171 4693 a FD(.)f(The)g(reason)109 4813 y(is)o(,)31
b(of)g(course)o(,)h(that)f(there)h(are)g(cancellations)g(betw)o(een)g
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(\223negativ)n(e\224)g(edges)o(.)f(Giv)n(en)109 5054
y Fh(x)g Fx(=)h(\()p Fy(x)399 5069 y Fr(1)438 5054 y
Fy(;)17 b(:)g(:)g(:)f(;)h(x)712 5069 y Fv(k)755 5054
y Fx(\))p FD(,)23 b(w)o(e)h(de\002ne)908 5256 y Ft(E)970
5215 y Fr(+)1028 5256 y Fx(\()p Fh(x)p Fx(\))k(:=)1328
5175 y Fw(\010)1386 5256 y Ft(f)p Fy(i;)17 b(j)6 b Ft(g)28
b(j)f Fx(1)g Ft(\024)i Fy(i)f(<)f(j)34 b Ft(\024)28 b
Fy(k)s(;)40 b(v)2385 5271 y Fv(ij)2446 5256 y Fx(\()p
Fh(x)p Fx(\))27 b Fy(>)h Fx(0)2768 5175 y Fw(\011)908
5410 y Ft(E)970 5369 y Fq(\000)1028 5410 y Fx(\()p Fh(x)p
Fx(\))g(:=)1328 5329 y Fw(\010)1386 5410 y Ft(f)p Fy(i;)17
b(j)6 b Ft(g)28 b(j)f Fx(1)g Ft(\024)i Fy(i)f(<)f(j)34
b Ft(\024)28 b Fy(k)s(;)40 b(v)2385 5425 y Fv(ij)2446
5410 y Fx(\()p Fh(x)p Fx(\))27 b Fy(<)h Fx(0)2768 5329
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432 y Fv(i)1031 417 y Ft(\000)e Fy(x)1185 432 y Fv(j)1222
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417 y Fy(;)17 b(:)g(:)g(:)f(;)h(x)3022 432 y Fv(k)3065
417 y Fx(\))p FD(,)872 605 y Ft(E)9 b Fx(\()p Fh(x)o
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b(j)6 b Ft(g)27 b(j)h Fx(1)f Ft(\024)h Fy(i)g(<)g(j)34
b Ft(\024)28 b Fy(k)s(;)40 b Ft(j)p Fy(x)2326 620 y Fv(i)2376
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Fy(f)1442 1248 y Fv(ij)1502 1233 y Fx(\()p Fh(x)p Fx(\))27
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(connected)i(con\002gu-)109 1807 y(r)o(ations)o(,)543
2031 y
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2205 y
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109
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3501 2693
a FD(\(25\))109 2991 y(where)c(the)f(sum)h(e)m(xtends)g(o)o(v)n(er)g(g)
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Ft(f)p Fx(1)p Fy(;)17 b(:)g(:)g(:)e(;)i(k)s Ft(g)23 b
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3224 y Fx(\))27 b(=)h(1)20 b(+)g Fy(f)2704 3239 y Fv(ij)2787
3224 y FD(and)k(e)m(xpand.)f(This)g(giv)n(es)109 3345
y(a)33 b(sum)g(o)o(v)n(er)h(g)o(r)o(aphs)o(.)g(The)f(g)o(r)o(aphs)i
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(side)f(of)109 3465 y(Eq.)24 b(\()288 3467 y
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288 3467
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2575 3465 a FD(\),)g(e)m(xcept)i
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y Fw(Q)2174 3615 y Fr(\()p Fv(ij)t Fr(\))p Fq(2)p Fv(\015)2394
3585 y Fy(f)2442 3600 y Fv(ij)2502 3585 y Fx(\()p Fh(x)p
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4320 y
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4701 y Fw(\014)1680 4761 y(\014)1680 4821 y(\014)1680
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(piece)n(wise)h(af\002ne)f(and)g(conca)n(v)n(e)h(in)f
Fy(\026)52 b Ft(2)353 4522 y Fx(\()p Ft(\0001)p Fy(;)17
b(e)657 4537 y Fq(1)731 4522 y Fx(])p FD(.)25 b(The)g(function)g
Fy(\027)37 b Ft(7!)30 b Fy(\026)p Fx(\()p Fy(\027)6 b
Fx(\))26 b FD(is)f(decreasing,)h(piece)n(wise)g(af\002ne)f(and)h(conca)
n(v)n(e)g(in)f Fy(\027)38 b Ft(2)353 4643 y Fx([0)p Fy(;)17
b Ft(1)p Fx(\))p FD(;)22 b(it)h(is)g(str)q(ictly)h(decreasing)h(in)e
Fy(\027)34 b Ft(2)28 b Fx([)p Fy(\027)1856 4607 y Fq(\003)1896
4643 y Fy(;)17 b Ft(1)p Fx(\))23 b FD(and)h(equals)g
Fy(\026)p Fx(\()p Fy(\027)6 b Fx(\))28 b(=)f Fy(e)2904
4658 y Fq(1)3003 4643 y FD(f)m(or)d Fy(\027)34 b Ft(\024)28
b Fy(\027)3364 4607 y Fq(\003)3404 4643 y FD(.)109 4709
y
SDict begin H.S end
109 4709 a 109 4709 a
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109 4709 a 109 4709 a
SDict begin [/View [/XYZ H.V]/Dest (Item.4) cvn /DEST pdfmark end
109 4709
a 0 TeXcolorgray 257 4846 a FD(2)p 0 TeXcolorgray 49
w(The)23 b(function)g Fy(\027)6 b Fx(\()p Fy(\026)p Fx(\))24
b FD(is)f(the)h(in)n(v)n(erse)g(of)f Fx([)p Fy(\027)1748
4810 y Fq(\003)1788 4846 y Fy(;)17 b Ft(1)p Fx(\))27
b Ft(3)h Fy(\027)35 b Ft(7!)27 b Fy(\026)p Fx(\()p Fy(\027)6
b Fx(\))p FD(:)23 b(f)m(or)h Fy(\026)j Ft(\024)h Fy(e)2893
4861 y Fq(1)2992 4846 y FD(and)c Fy(\027)34 b Ft(\025)28
b Fy(\027)3398 4810 y Fq(\003)3438 4846 y FD(,)1534 5065
y Fy(\027)35 b Fx(=)27 b Fy(\027)6 b Fx(\()p Fy(\026)p
Fx(\))28 b Ft(,)f Fy(\026)g Fx(=)h Fy(\026)p Fx(\()p
Fy(\027)6 b Fx(\))p Fy(:)109 5326 y FD(The)28 b(reciprocity)i(of)e
Fy(\026)p Fx(\()p Fy(\027)6 b Fx(\))29 b FD(and)g Fy(\027)6
b Fx(\()p Fy(\026)p Fx(\))28 b FD(is)h(analogous)h(to)e(the)g(equiv)n
(alence)i(of)e(the)g(g)o(r)o(and-canonical)j(and)109
5446 y(the)23 b(constant)g(pressure)h(ensemb)n(les)o(.)f(Indeed,)g(the)
g(pressure)i(and)e(the)g(Gib)o(bs)g(energy)h(\(per)g(par)s(ticle\))g
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109 129 a FD(both)29 b(obtained)g(as)h(Legendre)g(tr)o(ansf)m(or)r(ms)g
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514 y Fv(\032)1215 355 y Fw(\000)1261 435 y Fy(\026\032)22
b Ft(\000)h Fy(f)11 b Fx(\()p Fy(\014)6 b(;)17 b(\032)p
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Fv(v)2484 355 y Fw(\000)2550 409 y Fx(~)2529 435 y Fy(f)11
b Fx(\()p Fy(\014)6 b(;)17 b(v)t Fx(\))k(+)h Fy(pv)3039
355 y Fw(\001)3084 435 y Fy(;)109 703 y FD(with)302 677
y Fx(~)280 703 y Fy(f)11 b Fx(\()p Fy(\014)6 b(;)17 b(v)t
Fx(\))27 b(=)g Fy(v)t(f)11 b Fx(\()p Fy(\014)6 b(;)17
b(v)1005 667 y Fq(\000)p Fr(1)1098 703 y Fx(\))k FD(the)g(free)g
(energy)h(per)g(par)s(ticle)o(.)e(Equiv)n(alence)j(of)d(ensemb)n(les)i
(here)g(means)109 823 y(that)g Fy(p)p Fx(\()p Fy(\014)6
b(;)17 b Ft(\001)p Fx(\))23 b FD(and)h Fy(g)t Fx(\()p
Fy(\014)6 b(;)17 b Ft(\001)p Fx(\))k FD(are)j(reciprocal:)g(the)g(Gib)o
(bs)f(energy)i(is)e(the)h(same)f(as)h(the)g(chemical)f(potential.)109
991 y(Similar)q(ly)-8 b(,)24 b Fy(\026)p Fx(\()p Fy(\027)6
b Fx(\))25 b FD(looks)h(lik)n(e)f(a)h(Legendre)g(tr)o(ansf)m(or)r(m)f
(of)g Fy(k)34 b Ft(7!)c Fy(E)2277 1006 y Fv(k)2345 991
y FD(with)25 b(respect)h(to)e Fy(k)s FD(,)h(while)g Fy(\027)6
b Fx(\()p Fy(\026)p Fx(\))25 b FD(looks)109 1111 y(lik)n(e)33
b(a)f(Legendre)i(tr)o(ansf)m(or)r(m)f(of)f Fy(E)1301
1126 y Fv(k)1344 1111 y Fy(=k)j FD(with)e(respect)g(to)f
Fx(1)p Fy(=k)s FD(,)g(which)h(should)g(be)g(compared)g(with)f(the)109
1232 y(relations)23 b Fy(v)32 b Fx(=)27 b(1)p Fy(=\032)p
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109 1464 a Fu(Proof)24 b(of)f(Lemma)735 1464 y
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862 1464 a Fu(.)p 0 TeXcolorgray 48
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3097 1464 a FD(,)3154
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3201 1464 a FD(].)g(F)m(or)g
Fy(\027)6 b Fx(\()p Fy(\026)p Fx(\))p FD(,)109 1585 y(w)o(e)27
b(note)g(that)f(it)h(is)g(the)g(in\002m)o(um)f(of)h(a)g(f)m(amily)g(of)
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f(e)m(v)n(er)s(ywhere)i(diff)m(erentiab)n(le)o(,)d(with)h(der)q(iv)n
(ativ)n(e)h Ft(\000)p Fy(k)s Fx(\()p Fy(\026)p Fx(\))p
FD(,)f(the)109 1826 y(minimiz)o(er)22 b(of)h Fy(E)659
1841 y Fv(k)724 1826 y Ft(\000)g Fy(k)s(\026)p FD(.)g(In)g(par)s
(ticular)h Fy(k)s Fx(\()p Fy(\026)p Fx(\))j Ft(\025)i
Fx(1)p FD(,)22 b(hence)j Fy(\027)6 b Fx(\()p Fy(\026)p
Fx(\))23 b FD(is)h(str)q(ictly)g(decreasing.)109 1993
y(2.)18 b(W)m(e)h(pro)o(v)n(e)g(\223)p Ft(\))p FD(\224.)g(The)f(proof)h
(of)f(the)h(con)n(v)n(erse)h(is)f(similar)l(.)f(Thus)h(let)f
Fy(\026)27 b Ft(\024)h Fy(e)2641 2008 y Fq(1)2735 1993
y FD(and)19 b Fy(\027)34 b Fx(=)28 b(inf)3199 2008 y
Fv(k)3241 1993 y Fx(\()p Fy(E)3351 2008 y Fv(k)3397 1993
y Ft(\000)s Fy(k)s(\026)p Fx(\))p FD(.)109 2114 y(Clear)q(ly)-8
b(,)23 b Fy(\027)34 b Ft(\024)28 b Fx(inf)717 2129 y
Fv(k)760 2114 y Fx(\()p Fy(E)870 2129 y Fv(k)935 2114
y Ft(\000)23 b Fy(k)s(e)1134 2129 y Fq(1)1209 2114 y
Fx(\))k(=)h Fy(\027)1432 2077 y Fq(\003)1472 2114 y FD(,)23
b(and)h(f)m(or)f(e)m(v)n(er)s(y)i Fy(k)31 b Ft(2)d Fs(N)p
FD(,)1258 2347 y Fy(\027)34 b Ft(\024)28 b Fy(E)1517
2362 y Fv(k)1582 2347 y Ft(\000)23 b Fy(k)s(\026)51 b
Ft(\))27 b Fy(\026)g Ft(\024)2175 2280 y Fy(E)2247 2295
y Fv(k)2312 2280 y Ft(\000)22 b Fy(\027)p 2175 2324 292
4 v 2293 2416 a(k)2476 2347 y(;)109 2564 y FD(whence)h
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FD(.)22 b(On)g(the)g(other)h(hand,)f(if)g Fy(\026)27
b(<)h(e)1875 2579 y Fq(1)1950 2564 y FD(,)22 b(then)g
Fy(E)2255 2579 y Fv(k)2316 2564 y Ft(\000)d Fy(k)s(\026)27
b Ft(\025)h Fy(\027)2711 2528 y Fq(\003)2769 2564 y Fx(+)18
b Fy(k)s Fx(\()p Fy(e)3000 2579 y Fq(1)3093 2564 y Ft(\000)h
Fy(\026)p Fx(\))27 b Ft(!)g(1)c FD(as)109 2684 y Fy(k)30
b Ft(!)e(1)p FD(,)21 b(so)h(there)g(m)o(ust)g(be)g(a)g(\002nite)g
Fy(k)j FD(such)e(that)f Fy(\027)34 b Fx(=)27 b Fy(E)2078
2699 y Fv(k)2137 2684 y Ft(\000)16 b Fy(k)s(\026)p FD(.)22
b(It)g(f)m(ollo)o(ws)g(that)f Fy(\026)27 b Fx(=)h(\()p
Fy(E)3199 2699 y Fv(k)3258 2684 y Ft(\000)16 b Fy(\027)6
b Fx(\))p Fy(=k)31 b Ft(\025)109 2805 y Fy(\026)p Fx(\()p
Fy(\027)6 b Fx(\))p FD(,)32 b(whence)h Fy(\026)46 b Fx(=)f
Fy(\026)p Fx(\()p Fy(\027)6 b Fx(\))p FD(.)32 b(If)g
Fy(\026)45 b Fx(=)h Fy(e)1500 2820 y Fq(1)1575 2805 y
FD(,)32 b(then)g Fy(\027)52 b Fx(=)46 b Fy(\027)2104
2768 y Fq(\003)2176 2805 y FD(and)33 b(the)f(claim)g(f)m(ollo)o(ws)h
(from)f(the)g(gener)o(al)109 2925 y(inequality)23 b Fy(\026)p
Fx(\()p Fy(\027)6 b Fx(\))28 b Ft(\024)g Fy(e)857 2940
y Fq(1)932 2925 y FD(.)p 3590 2925 4 58 v 3594 2871 51
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3303 225 4 v 1694 3394 a Fy(k)25 b Ft(\000)e Fx(1)1928
3326 y Fy(;)102 b(\027)2105 3341 y Fr(1)2172 3326 y Fx(=)28
b Ft(\000)p Fy(\026)2412 3341 y Fr(1)2451 3326 y Fy(:)109
3543 y FD(Then)p 0 TeXcolorgray 1 0.6 0 0 TeXcolorcmyk
233 3764 a Fj(\004)p 0 TeXcolorgray 0 TeXcolorgray 49
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Fy(e)862 3779 y Fq(1)961 3764 y FD(and)c Fy(\027)1179
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817 4190 a 24 w FD(implies)23
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4190 y Ft(\024)29 b Fy(e)2154 4205 y Fq(1)2252 4190 y
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Fy(\027)2693 4154 y Fq(\003)2733 4190 y FD(.)24 b(Moreo)o(v)n(er)l(,)f
(b)n(y)i(de\002nition,)109 4311 y Fy(\027)163 4275 y
Fq(\003)230 4311 y Ft(\024)j Fy(E)407 4326 y Fr(1)469
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b Ft(\000)p Fy(e)942 4326 y Fq(1)1041 4311 y FD(and)1116
4535 y Fy(\027)1164 4550 y Fr(1)1231 4535 y Fx(=)g(sup)1389
4617 y Fv(k)1563 4467 y Fy(E)1635 4482 y Fv(k)p 1508
4512 225 4 v 1508 4603 a Fx(1)22 b Ft(\000)h Fy(k)1770
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4467 y Fy(E)2209 4482 y Fv(k)p 2082 4512 V 2082 4603
a Fx(1)22 b Ft(\000)g Fy(k)2344 4535 y Fx(=)27 b Ft(\000)p
Fy(e)2569 4550 y Fq(1)109 4769 y FD(so)36 b(that)g Fy(\027)466
4733 y Fq(\003)559 4769 y Ft(\024)53 b(\000)p Fy(e)811
4784 y Fq(1)939 4769 y Ft(\024)h Fy(\027)1118 4784 y
Fr(1)1157 4769 y FD(.)36 b(If)g(in)g(addition)g Fy(\026)1795
4784 y Fr(1)1887 4769 y Fx(=)52 b Fy(e)2060 4784 y Fq(1)2135
4769 y FD(,)36 b(then)g Fy(\027)2444 4784 y Fr(1)2537
4769 y Fx(=)52 b Ft(\000)p Fy(e)2787 4784 y Fq(1)2899
4769 y FD(and)37 b(f)m(or)f(all)g Fy(k)56 b Ft(2)d Fs(N)p
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b Fx(\()p Fy(k)33 b Ft(\000)d Fx(1\))p Fy(e)756 4905
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y Ft(\000)31 b Fy(k)s(e)2422 4905 y Fq(1)2497 4890 y
Fx(\))46 b Ft(\025)i(\000)p Fy(e)2828 4905 y Fq(1)2903
4890 y FD(.)33 b(Since)h(in)g(an)o(y)f(case)109 5010
y Fy(\027)163 4974 y Fq(\003)230 5010 y Ft(\024)28 b
Fy(e)380 5025 y Fq(1)455 5010 y FD(,)23 b(w)o(e)h(get)g
Fy(\027)829 4974 y Fq(\003)896 5010 y Fx(=)k Fy(e)1045
5025 y Fq(1)1120 5010 y FD(.)109 5178 y(If)c Fy(\026)240
5193 y Fr(1)309 5178 y Fy(<)30 b(e)460 5193 y Fq(1)535
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y Fq(1)2888 5418 y Fy(:)p 3590 5418 4 58 v 3594 5365
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